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I was just philosophizing about the philosophy of mathematics. Then at one point I philosophized: is there a philosophy of philosophy? Is that meta-philosophy, or is that just philosophy again? Can someone philosophize with me about this matter of the highest importance? I cannot philosophize this deeply without a philosophic consult.

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    Go for a doctorate and see what happens! – elliot svensson Dec 17 '18 at 19:47
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    It is called metaphilosophy. There is an article about it in Wikipedia. Could you make your question more precise? – Conifold Dec 17 '18 at 22:00
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    What's the purpose of phrasing your question in this way instead of just asking a concise question? Does it do anything to help get a better quality answer? I feel like at best it's just distracting and at worst its purposefully obtuse and off-putting. Like, everything other than "is there a philosophy of philosophy? Is that meta-philosophy, or is that just philosophy again?" adds nothing to your question. – Not_Here Dec 18 '18 at 1:43
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    No more than the history of history, which makes perfect sense. Thucydides was an ancient historian and John Lukacs is a modern one. That's the history of history. And for that matter we have metamathematics, using mathematical techniques to study the nature of mathematics. Not every self-reference is a paradox. Now I'm going to go read a book on bibliology, the study of books. – user4894 Dec 18 '18 at 3:21
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It is certainly one view. The most compact statement of the problem is Munchausen's trilemma. All mathematics and logic fall at this, being based on axioms, infinite regress, or circular reasoning, none of which can justify the choice of the method itself (see also the problem of induction). This can be interpreted as the result of Godel-incompleteness, that in any reasonably complex system or language there are true statements which are unprovable within the system.

A proposed solution is Strange Loops, where the trilemma is stepped around through 'tangled hierarchies'; the appeal of patterns and structures themselves, and use of the capacity that seems essential to consciousness of stepping outside of any given system and asking 'why?', then linking together of systems in tangles.

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Here's a quote from Wikipedia:

Metaphilosophy (sometimes called philosophy of philosophy) is "the investigation of the nature of philosophy". Its subject matter includes the aims of philosophy, the boundaries of philosophy, and its methods. Thus, while philosophy characteristically inquires into the nature of being, the reality of objects, the possibility of knowledge, the nature of truth, and so on, metaphilosophy is the self-reflective inquiry into the nature, aims, and methods of the activity that makes these kinds of inquiries, by asking what is philosophy itself, what sorts of questions it should ask, how it might pose and answer them, and what it can achieve in doing so. It is considered by some to be a subject prior and preparatory to philosophy, while others see it as inherently a part of philosophy, or automatically a part of philosophy while others adopt some combination of these views.

Some philosophers consider metaphilosophy to be a subject apart from philosophy, above or beyond it, while others object to that idea. Timothy Williamson argues that the philosophy of philosophy is "automatically part of philosophy", as is the philosophy of anything else.

Let's verify the implications of the usage--'Philosophy (2nd)....of....Philosophy (1st).

Directly or indirectly this implies the first philosophy is imperfect. So, if you use this kind of a term you imply the imperfection of Philosophy. Remember, the 2nd also is Philosophy. Since thought is the main thing involved in Philosophy, it can never rise above the level of the term, 'Philosophy'. When you think of Philosophy of Philosophy, YOUR MIND BECOMES DISTURBED for Philosophy is always related to mind. It can't take you anywhere if your aim is to find out the truth behind philosophical thoughts. Such thoughts stop eventually.

We can't see well under water if it is disturbed. Similarly, when our mind is disturbed we can't realize the Truth. All these imply that WE CAN'T MAKE USE OF PHILOSOPHY OF PHILOSOPHY AS A NEW BRANCH. When I went through the details of Philosophy of Philosophy I could understand that all the description regarding it, is a normal thing that good philosophers do before taking a decision. Though Philosophy of Philosophy is "the investigation of the nature of philosophy" it can't evaluate most philosophers/philosophy correctly. Many philosophers who deals this subject would be at logger heads. So IMHO, in course of time this will dissolve in Philosophy.

To avoid this, we'd better accept the words of Ramana Maharshi.

One morning Sri Bhagavan quoted from a journal the following sentence: "Where psychology ends, philosophy begins" and added his own remark, "Where philosophy ends spirituality begins."

For details please go through the link:

http://www.arunachala.org/newsletters/1992/jan-feb

Almost in all cases we can't make use of the same subject for finding out the truth behind that subject.

For your main question,

Does the snake bite its own tail?

you could make use of the following link:

Can a tool use itself?

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